A sequent root-first proof-search procedure for intuitionistic propositional logic is presented. The procedure is obtained from modified intuitionistic multi-succedent and classical sequent calculi, making use of Glivenko’s Theorem. We prove that a sequent is derivable in a standard intuitionistic multi-succedent calculus if and only if the corresponding prefixed-sequent is derivable in the procedure.

Equational reasoning arises in many areas of mathematics and computer science. It is a cornerstone of algebraic reasoning and results essential in tasks of specification and verification in functional programming, where a program is mainly a set of equations. The usual manipulation of identities while conducting informal proofs obviates many intermediate steps that are neccesary while developing them using a formal system, such as the equationally complete Birkhoff calculus ${\mathcal{B}}$. This deductive system does not fit in the common manner of (...) doing mathematical proofs, and it is not compatible with the mechanisms of proof assistants. The aim of this work is to provide a deductive system ${\mathcal{B}}^{\textrm{GOAL}}$ for equality, equivalent to ${\mathcal{B}}$ but suitable for constructing equational proofs in a backward fashion. This feature makes it adequate for interactive proof-search in the approach of proof assistants. This will be achieved by turning ${\mathcal{B}}^{\textrm{GOAL}}$ into a transition system of formal tactics in the style of Edinburgh LCF, such transformation allows us to give a direct formal definition of backward proof in equational logic. (shrink)

A survey of works on automatic theorem-proving in the ussr 1964-1970. the philosophical problems are not touched.

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical (...) logic, and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

The reductive, as opposed to deductive, view of logic is the form of logic that is, perhaps, most widely employed in practical reasoning. In particular, it is the basis of logic programming. Here, building on the idea of uniform proof in reductive logic, we give a treatment of logic programming for BI, the logic of bunched implications, giving both operational and denotational semantics, together with soundness and completeness theorems, all couched in terms of the resource interpretation of BI’s semantics. (...) We use this set-up as a basis for exploring how coalgebraic semantics can, in contrast to the basic denotational semantics, be used to describe the concrete operational choices that are an essential part of proof-search. The overall aim, toward which this paper can be seen as an initial step, is to develop a uniform, generic, mathematical framework for understanding the relationship between the deductive structure of logics and the control structures of the corresponding reductive paradigm. (shrink)

The cut-free Gentzen-type sequent calculus LLK for the logic of likelihood is introduced in the paper. It is proved that the calculus is sound and complete for LL. Using the introduced calculus LLK, a decision procedure for LL is presented.

In this paper, we consider three different search strategies for a cut-free sequent system formalizing orthologic, and estimate the respective search spaces. Applying backward search, there are classes of formulae for which both the minimal proof length and the search space are exponential. In a combined forward and backward approach, all proofs are polynomial, but the potential search space remains exponential. Using a forward strategy, the potential search space becomes polynomial yielding a polynomial (...) decision procedure for orthologic and the word problem for free ortholattices. (shrink)

We propose a new sequent calculus for bi intuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cut elimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cut elimination proof. We then present the derived calculus, and (...) then present a proof search strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for bi intuitionistic logic. As far as we know, our new calculus is the first sequent calculus for bi intuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cut elimination proof, and which can be used naturally for backwards proof search. Keywords: Bi-intuitionistic logic, display calculi, proof search. (shrink)

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical (...) logic, and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism, Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search including proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences.

In this paper, we consider three different search strategies for a cut-free sequent system formalizing orthologic, and estimate the respective search spaces. Applying backward search, there are classes of formulae for which both the minimal proof length and the search space are exponential. In a combined forward and backward approach, all proofs are polynomial, but the potential search space remains exponential. Using a forward strategy, the potential search space becomes polynomial yielding a polynomial (...) decision procedure for orthologic and the word problem for free ortholattices. (shrink)

. This paper continues a systematic approach to build natural deduction calculi and corresponding proof procedures for non-classical logics. Our attention is now paid to the framework of paraconsistent logics. These logics are used, in particular, for reasoning about systems where paradoxes do not lead to the `deductive explosion', i.e., where formulae of the type `A follows from false', for any A, are not valid. We formulate the natural deduction system for the logic PCont, explain its main concepts, define (...) a proof searching technique and illustrate it by examples. The presentation is accompanied by demonstrating the correctness of these developments. (shrink)

We study from the proof complexity perspective the proof search problem : •Is there an optimal way to search for propositional proofs?We note that, as a consequence of Levin’s universal search, for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal proof search algorithm exists without restricting proof systems iff a p-optimal proof (...) system exists.To characterize precisely the time proof search algorithms need for individual formulas we introduce a new proof complexity measure based on algorithmic information concepts. In particular, to a proof system P we attach information-efficiency function $i_P$ assigning to a tautology a natural number, and we show that: • $i_P$ characterizes time any P-proof search algorithm has to use on $\tau $,•for a fixed P there is such an information-optimal algorithm,•a proof system is information-efficiency optimal iff it is p-optimal,•for non-automatizable systems P there are formulas $\tau $ with short proofs but having large information measure $i_P$.We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P$ where no super-polynomial size lower bounds are known. We also point out connections of the new measure with some topics in proof complexity other than proof search. (shrink)

ABSTRACT This paper presents an attempt to create a shell for generic interactive proof search and proof assistant software based on it.

The Carnegie Mellon Proof Tutor project was motivated by pedagogical concerns: we wanted to use a "mechanical" (i.e. computerized) tutor for teaching students..

Wilfred Sieg and Clinton Field. Automated Search for Gödel's Proofs.

Wilfred Sieg. Mechanisms and Search: Aspects of Proof Theory.

From Kant’s first published work to recent articles in the physics literature, philosophers and physicists have long sought an answer to the question, why does space have three dimensions. In this paper, I will flesh out Kant’s claim with a brief detour through Gauss’ law. I then describe Büchel’s version of the common argument that stable orbits are possible only if space is three-dimensional. After examining objections by Russell and van Fraassen, I develop three original criticisms of my own. These (...) criticisms are relevant to both historical and contemporary proofs of the dimensionality of space (in particular, a recent one by Burgbacher, F. Lämmerzahl, C., and Macias). In general I argue that modern “proofs” of the dimensionality of space have gone off track. (shrink)

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame (...) class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope. (shrink)

This work provides proof-search algorithms and automated counter-model extraction for a class of STIT logics. With this, we answer an open problem concerning syntactic decision procedures and cut-free calculi for STIT logics. A new class of cut-free complete labelled sequent calculi G3LdmL^m_n, for multi-agent STIT with at most n-many choices, is introduced. We refine the calculi G3LdmL^m_n through the use of propagation rules and demonstrate the admissibility of their structural rules, resulting in auxiliary calculi Ldm^m_nL. In the single-agent (...) case, we show that the refined calculi Ldm^m_nL derive theorems within a restricted class of (forestlike) sequents, allowing us to provide proof-search algorithms that decide single-agent STIT logics. We prove that the proof-search algorithms are correct and terminate. (shrink)

In his warranted christian belief, Alvin Plantinga launches a forceful attack on apophaticism, the view that God is in some sense or other beyond description. This paper explores his attack before searching for a Plantinga-proof formulation of apophaticism.

A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.

This paper studies proof systems for the logics of super-strict implication ST2–ST5, which correspond to C.I. Lewis’ systems S2–S5 freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating STn in Sn and backsimulating Sn in STn, respectively(for n=2,...,5). Next, G3-style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of G3-style calculi, that they are sound and complete, (...) and it is shown that the proof search for G3. ST2 is terminating and therefore the logic is decidable. (shrink)

A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich (...) contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness. (shrink)

Admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In this paper, a Gentzen-style framework is introduced for analytic proof systems that derive admissible rules of non-classical logics. While Gentzen systems for derivability treat sequents as basic objects, for admissibility, the basic objects are sequent rules. Proof systems are defined here for admissible rules of classes of modal logics, including K4, S4, and GL, and also Intuitionistic Logic IPC. (...) With minor restrictions, proof search in these systems terminates, giving decision procedures for admissibility in the logics. (shrink)

There is no topic that attracts more attention, more energy and time and devotion, than love. As long as there's been recorded history, love has taken center seat as the inspiration for countless paintings, instigator of wars, muse of untold poets and musicians. And just as poetry, art and music have the ability to communicate something about love that is difficult to articulate with words, the same is true of mathematics. Of course, mathematics can't easily help us translate the emotional (...) side of love, emotions rarely behave in a neatly ordered, rational and easily predictable way. It is difficult to quantify the rollercoaster of romance or to define how lovers might feel via a set of simple equations. But that doesn't mean that mathematics isn't crucial to understanding love. Love, like most things in life, is full of patterns. And mathematics is ultimately the study of patterns, from predicting the weather to the fluctuations of the stock market, the movement of planets or the growth of cities. These patterns twist and turn and warp and evolve just as the rituals of love do. In [this book, the author] takes the listener on a fascinating journey through the patterns that define our love lives, tackling some of the most common yet complex questions pertaining to love: What's the chance of us finding love? What's the chance that it will last? How does online dating work, exactly? When should you settle down? How can you avoid divorce? When is it right to compromise? Can game theory help us decide whether or not to call? From evaluating the best strategies for online dating to defining the nebulous concept of beauty, Dr. Fry proves, with great insight, wit and fun, that math is a surprisingly useful tool to negotiate the complicated, often baffling, sometimes infuriating, always interesting, patterns of love. (shrink)

Traditional frameworks for evaluating scientific models have tended to downplay their exploratory function; instead they emphasize how models are inherently intended for specific phenomena and are to be judged by their ability to predict, reproduce, or explain empirical observations. By contrast, this paper argues that exploration should stand alongside explanation, prediction, and representation as a core function of scientific models. Thus, models often serve as starting points for future inquiry, as proofs of principle, as sources of potential explanations, and as (...) a tool for reassessing the suitability of the target system. This is illustrated by a case study of the varied career of reaction-diffusion models in the study of biological pattern formation, which was initiated by Alan Turing in a classic 1952 paper. Initially regarded as mathematically elegant, but biologically irrelevant, demonstrations of how, in principle, spontaneous pattern formation could occur in an organism, such Turing models have only recently rebounded, thanks to advances in experimental techniques and computational methods. The long-delayed vindication of Turing’s initial model, it is argued, is best explained by recognizing it as an exploratory tool. (shrink)

This paper studies proof systems for the logics of super-strict implication \(\textsf{ST2}\) – \(\textsf{ST5}\), which correspond to C.I. Lewis’ systems \(\textsf{S2}\) – \(\textsf{S5}\) freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating \(\textsf{STn}\) in \(\textsf{Sn}\) and backsimulating \(\textsf{Sn}\) in \(\textsf{STn}\), respectively (for \({\textsf{n}} =2, \ldots, 5\) ). Next, \(\textsf{G3}\) -style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are (...) distinctive of \(\textsf{G3}\) -style calculi, that they are sound and complete, and it is shown that the proof search for \(\mathsf {G3.ST2}\) is terminating and therefore the logic is decidable. (shrink)

In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. (...) We then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91]. (shrink)

This paper is a continuation of dag-like proof compression research initiated in [9]. We investigate proof compression phenomenon in a particular, most transparent case of propositional DNF Logic. We define and analyze a very efficient semi-analytic sequent calculus SEQ*0 for propositional DNF. The efficiency is achieved by adding two special rules CQ and CS; the latter rule is a variant of the weakened substitution rule WS from [9], while the former one being specially designed for DNF sequents. We (...) show that dag-like SEQ*0 has good deterministic proof search strategy. Furthermore, we expose six examples showing that dag-like deducibility in SEQ*0 admits exponential-size proof compression, as compared to familiar proof systems like cutfree sequent calculi, resolution and cutting plane proof systems. (shrink)

Goal Directed Proof Theory presents a uniform and coherent methodology for automated deduction in non-classical logics, the relevance of which to computer science is now widely acknowledged. The methodology is based on goal-directed provability. It is a generalization of the logic programming style of deduction, and it is particularly favourable for proof search. The methodology is applied for the first time in a uniform way to a wide range of non-classical systems, covering intuitionistic, intermediate, modal and substructural (...) logics. The book can also be used as an introduction to these logical systems form a procedural perspective. Readership: Computer scientists, mathematicians and philosophers, and anyone interested in the automation of reasoning based on non-classical logics. The book is suitable for self study, its only prerequisite being some elementary knowledge of logic and proof theory. (shrink)

Cook and Reckhow [5] pointed out that $\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$ iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can (...) be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:•There exists a p-time function g stretching each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {N}\mathcal {P}$ sets.We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from [18] is a good candidate for g. We define a new hardness property of generators, the $\bigvee $ -hardness, and show that one specific gadget generator is the $\bigvee $ -hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite $\mathcal {N}\mathcal {P}$ sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite $\mathcal {N}\mathcal {P}$ sets. (shrink)